#180568

order	512
nilpotency class	3
isoclinism	isoclinic class of #180005
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(2,6)(3,4)(7,8)(9,10,14,18)(11,19,22,24)(12,16,13,17)(15,20,23,21)(25,26)(27,30)(28,36)(29,33)(31,35)(32,37)(34,39)(38,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,24)(18,23)(25,27,30,26)(28,34,38,32)(29,33,31,35)(36,37,40,39);(3,4)(6,7)(9,11)(10,15)(12,21)(13,20)(14,22)(16,24)(17,19)(18,23)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,40)(35,36);(1,3,5,4)(2,7,8,6)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,19,23,24)(25,29,30,31)(26,33,27,35)(28,37,38,39)(32,40,34,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,d^b=dgi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 87764);